The progressive cavity pump—also referred to hereinafter by the abbreviation PCP—was invented by Rene Moineau in 1930 and current industrial PCPs correspond to the basic principles.
To describe the architecture of the PCP according to the invention, we begin by showing the operation of the conventional PCP while emphasizing the processes that affect the reliability and performance of this pump.
We then present the PCP according to the invention, as well as its operation and its ability to improve reliability and performance.
The architecture of the conventional PCP includes a metal helical rotor inside a helical stator that is elastic (of elastomer) or rigid (metal, of composite materials).
FIG. 2A shows a longitudinal cross-section of a conventional PCP with an elastic helical stator, according to the prior art. FIG. 2B shows an enlarged view of the boxed area B indicated in FIG. 2A.
As can be seen in FIGS. 2A and 2B, the conventional PCP 1 with elastic stator consists of a metal helical rotor 2 rotating within a helical stator 3, usually of elastomer, contained in a casing 5. The geometry of the PCP results in a set of isolated cavities 4 of constant volume, defined between the rotor 2 and the stator 3, which the rotor 2 displaces from the intake or inlet (low pressure) toward the discharge or outlet (high pressure).
In this sense, the PCP is a positive displacement pump capable of transporting various products: more or less viscous liquids, multiphase mixtures (liquid, gas, solid particles).
The stator 3, of elastomer, has radial thickness H1 in its concave portions and radial thickness H2 in its convex portions. For example, the stator 3 having an outer diameter of 7 cm has thicknesses of 2.5 for H1 and 1.5 for H2.
To ensure that the PCP 1 compresses the fluids (liquids and gases) with a virtually fluidtight seal between the cavities 4, the rotor 2 with its helical rotation exerts a compressive force on the elastomer of the stator 3. Given the risk of damage to the stator 3, the reliability of PCPs is a major issue in the industrial application of these pumps.
For example, the oil industry uses PCPs in deep wells to pump mixtures of oil, water, and gas, that are carrying solid particles. In the pumping conditions downhole, the elastomer of the stator 3, subjected to complex thermal, chemical, and mechanical processes (dynamic forces and pressure), expands and thus increases the forces exerted by the rotor 2 on the stator 3.
The service life of conventional PCPs is therefore considerably reduced.
Using the diagrams in FIGS. 2A and 2B, we can describe the behavior of the stator 3 of the conventional PCP subjected to the forces exerted by the rotor 2 in its helical motion.
The operation of the conventional PCP 1 includes close contact, by the interference fit between the rotor 2 and the elastomeric stator 3, which ensures two combined functions:                providing the relative fluidtightness necessary for pumping the cavities 4, from intake (low pressure) to discharge (high pressure),        concentrating and transmitting forces through the stator 3 to the casing 5.        
Thus, to minimize leakage between the cavities 4, the rotor 2 exerts compressive force P1 on the stator 3, which deforms by a height h1, generally called the interference, along an interference length L1. In the case mentioned above, this length L1 is about 4 cm.
As a result, the interference h1 between the rotor 2 and the stator 3 provides virtually fluidtight cavities 4, thus limiting leakage.
At the same time, the helical motion of the rotor 2 generates a shear force Q1 on the stator 3. The greater the interference h1, the greater the compressive forces P1 and shear forces Q1, and the greater the risk of damage to the stator 3.
In practice, an initial interference h1 between the rotor 2 and the stator 3 is adopted; this is the result of a compromise between acceptable stresses and a relative fluidtightness to limit leaks. For example, for the abovementioned stator 3 having an outside diameter of 7 cm, an initial interference h1 of 0.5 mm is adopted.
However, in the downhole conditions of an oil well, the stator 3 undergoes changes leading to an increase in the thicknesses H1 and H2 of the stator 3 and in the interference h1 between the rotor 2 and the stator 3.
Several phenomena can lead to the increase in the thicknesses H1 and H2 of the stator 3 and in the interference h1.                First, thermodynamic processes lead to expansion of the stator 3. In particular:                    the petroleum products downhole often have high temperatures,            gas compression in the PCP causes the temperature to rise, particularly in the portion near the pump outlet (high pressure),            the friction between the rotor 2 and the stator 3 also leads to an increase in temperature,            the large thickness H1 of the stator 3 limits the dissipation of heat to the outside, further contributing to the expansion of the stator 3.                        The chemical reaction of the elastomer of the stator 3 with the pumped fluids (liquids and gases) often causes the stator 3 to swell.        Due to pressure in the pump, the presence of gas leads to swelling of the stator 3; in effect, the pressurized gas penetrates the elastomer and acts on the stator 3 during pressure variations in the pump.        Lastly, the helical motion and vibrations of the rotor 2 generate dynamic forces on the stator 3 as a function of the interference h1 among other factors.        
Under these conditions, the interference h1 is the determining factor in the balance between fluidtightness and the contact forces between the rotor 2 and stator 3.
Analysis of the impact of the interference h1 on the compressive P1 and shear Q1 forces shows the risk of damage to the stator 3.
For this, we adopt the following notation:                E, the elastic modulus of the elastomer (the stator 3)        R, the radius of the rotor 2 (FIG. 2A)        C, the constants        V, the rotational speed of the rotor 2 (revolutions/minute).        
In general, the functions f(V) are used to indicate the influence of the rotational speed V of the rotor 2 on the compressive P1 and shear Q1 forces, and on the interference h1 between the rotor 2 and stator 3.
The analytical formulation thus demonstrates the correlation between the interference h1 and the compressive P1 and shear Q1 forces; to facilitate interpretation, the other parameters are grouped together.
As can be seen in FIG. 2B, the compressive force P1 applied by the rotor 2 results in the interference h1 with the stator 3.
The viscoelastic model (Bowden) yields expressions (1, 2), linking the compressive force P1 and interference h1:P1=C1.f1(V).h13/2.E.R1/2  (1)h1=C2.f2(V).(P1/E)2/3.R−1/3  (2)
Linear approximation (elastic model, Boussinesq) facilitates the interpretation (expressions 3, 4):P1=C3.f3(V).h1.R.E  (3)h1=C4.f4(V).P1/(R.E)  (4)
These relations (1,2,3,4) show that the compressive forces P1 are significant when the interference h1 is large. In addition, these forces are concentrated within a volume in the immediate vicinity of the contact surface S1 (FIG. 2B).
Therefore, the swelling of the stator 3 increases interference h1 and leads to significant compressive forces P1 concentrated at the contact surfaces S1. These contact surfaces S1, shown in FIGS. 2A and 2B, are surfaces of the inner face of the elastomer of the stator 3, positioned opposite a convex portion of the rotor 2.
Relations (3,4) describe the elastic compression (Boussinesq) of the stator 3 as a result of compressive forces P1. If we denote the stiffness of the elastomeric stator as Ks, we find that the behavior of the stator 3 is equivalent to the response of an elastic spring at constant rotor speed V:P1=Ks.h1h1=P1/Ks Ks=C3.E.R  (5)
The helical motion of the rotor 2 generates a shear force Q1 which also depends on the interference h1 (FIG. 2B). The elastoplastic approach (Hill) leads to the relation:Q1=C5.f5(V).h1(1−h1/H1).E.R  (6)
The shear forces Q1 exerted on the stator 3 are a function of the interference h1, Q1=F(h1); the greater the interference h1, the greater the risk of damaging the stator. However, as previously mentioned, conventional PCPs 1 must have an initial interference h1 of around 0.5 mm to ensure fluidtight cavities 4. Given the production conditions in oil wells (thermodynamic-chemical-dynamic), the stator undergoes an increase in thicknesses H1 and H2 of about 5 to 10%, and, depending on the properties of the elastomer, the interference increases by about 1 mm which means that it is multiplied by 2. Under these conditions, the compressive P1 and shear Q1 forces are also multiplied by 2. As for the dynamic forces exerted by the helical rotation of the rotor 2 on the stator 3, these depend on the rotational speed V of the pump; in order to produce (flows and pressures) under cost-effective conditions, PCPs rotate at the speed of 200-500 rpm. Given the pumping conditions in the well, the service life of an elastomeric stator 3 is significantly reduced; experience shows that the average is one year, but damaged stators have been observed after 1-3 months of operation.
The vibrations of the rotor 2 depend on the natural frequency of the rotor 2 and on the rotational speed of the pump and can be very significant, especially in the resonance between the rotor 2 and the rotational speed (frequency). The amplitude of the vibrations of the rotor 2, perpendicular to the axis X-X, creates increased interference h1, and therefore the compressive P1 and shear Q1 forces exerted on the stator 3 are also increased.
Thus, the operation of conventional PCPs 1 focuses the forces at the contact between the rotor 2 and stator 3, and often leads to degradation of the stator 3. In practice, the oil company must remove the damaged pump from the well and replace it; this is a long process, during which the well is no longer in production and therefore there is a significant economic impact.
The more recent PCP 24 which includes a rigid helical stator (metal, composite materials) is shown in longitudinal section in FIG. 6. This pump 7 comprises a helical rotor which rotates inside the rigid helical stator 25; there is some clearance 26 between the rotor 7 and the stator 25.
In practice, the stator 25 made of a rigid material (metal, composite materials) is mounted inside the casing 19; then, the helical rotor 7 is inserted inside the rigid stator 25, with some clearance 26. The architecture of this PCP is similar to that of a conventional PCP: the difference lies in the fact that there is clearance 26 between the rotor 7 and the rigid stator 25.
This PCP 24 is used in particular for pumping viscous liquids (heavy oils); the rotor 7 carries the viscous liquid and a liquid film is formed in the clearance 26 between the rotor 7 and the rigid stator 25. Depending on the manufacturing method, this clearance is less than 1 mm.
As a result, with no contact between the rotor 7 and the stator 25, the PCP 24 pumps the viscous liquids (heavy oils).
Given the existence of this clearance 26, the rotation of the helical rotor 7 at speeds of 200-500 rpm generates vibrations (resonance, unstable vibrations) and shocks between the rotor 7 and the stator 25.
For example, if the natural frequencies of the rotor 7 and/or the rigid stator 25 are of the same order of magnitude as that of the rotational speed (speed of 200-500 revolutions per minute), the forces due to vibrations and shocks are multiplied by 6-8; the rotor 7 and the stator 25 cannot resist these forces for very long.
The dynamic response of the PCP 24 with rigid stator 25 can damage the rotor 7 and/or stator 25. In these circumstances, the oil company must replace the pump, which is a major operation with significant economic impact.